New Exponential operators connected with a^2+x^2: a generalization of Post-Widder and Ismail May

TL;DR

This paper introduces a new exponential operator linked with a^2+x^2, providing asymptotic approximation estimates, graphical convergence analysis, and comparisons with existing operators like Post-Widder and Ismail-May.

Contribution

It presents a novel exponential operator generalizing known operators, with detailed asymptotic analysis, convergence visualization, and optimal parameter determination.

Findings

Operator converges to target functions graphically.

Asymptotic approximation formulas are established.

Optimal parameter a maximizes approximation quality.

Abstract

The present study offers a general exponential operator connected with a^2+x^2; for positive real "a". We estimate the asymptotic formula for simultaneous and ordinary approximation of the constructed operator. In the last section, we graphically interpret the created operator's convergence to two periodic functions "x sin(x)" and "-x/2*cos(pi*x)". We also consider the limiting case a tends to 0; which provides Post-Widder operator. In addition, we analyze each particular case of the defined operator and determine the optimal value of "a", that would yield the greatest approximation; this facilitates us to contrast the well-known operators existing in the literature, especially the Post-Widder operator and the operator due to Ismail-May.